Pre-Calculus Assignment Solution - Module Name - Semester 1

This document presents a complete solution to a Pre-Calculus assignment. The solution covers various topics including trigonometric equations, trigonometric functions, and complex numbers. The first section of the assignment focuses on solving trigonometric equations, demonstrating the application of trigonometric identities. The second section delves into the analysis of trigonometric functions, including amplitude, period, and phase shifts. The third section involves operations with complex numbers in polar form, including multiplication and division. The final section converts complex numbers from polar to rectangular form. This assignment provides a comprehensive overview of key concepts in Pre-Calculus and is designed to assist students in understanding and mastering the subject matter. The student's work is available on Desklib, a platform offering study resources and solutions to help students excel in their coursework.

Running head: Pre-calculus
Pre Calculus
Name of the Student
Name of the University

2Pre-Calculus
Table of Contents
Q1...............................................................................................................................................3
Q2...............................................................................................................................................3
Q3...............................................................................................................................................4
Q4...............................................................................................................................................4

3Pre-Calculus
Q1.
a)
2cos2(x) + 1 = 0.
2cos2(x)-1 +2 = 0.
Cos2x = -2.
But −1 ≤ cosθ ≤1, therefore the equation has no solutions.
b)
tan2(x) + 3tanx + 2 = 0.
= tan2(x) + 2tan(x)+ tan(x) +2 = 0
= tan(x)( tan(x)+2) + 1(tanx+2) = 0
= (tan(x) +1)(tanx+2) = 0
∴ tanx=−1∨tanx=−2
∴ x=−π
4 , tan−1 (−2 ) .
Q2.
a)
y = 7sin (1/3x + π/2 )
Amplitude = 7
Period =
2 π
1
3
=6 π .
Phase Shift = π
2
When, x = π
6
y = 7sin (
1
3∗π
6
+ π/2 )
= 7sin( 10 + π/2)
= 7cos10.
b)

4Pre-Calculus
y = -3cos(4/5x - π)
Amplitude: 3
Period:
2 π
4
5
= 5 π
2
Phase shift = −π
When x = π / 6
y = -3cos(
4
5∗π
6
- π)
= -3cos (24 - π)
= -3cos (π- 24)
= 3cos24.
Q3.
a)
4( cos 80˚ + i sin 80˚) * 3(cos 40˚ + i sin 40˚)
= 12( cos(80+40) + i sin(80+40) )
= 12 (cos120 + i sin120)
= 12 (cos(π -60) + i sin (π-60) )
= 12 ( -1/2 + i √3
2 )
= -6 + 6 √3
b)
15(cos 5π/6 + i sin 5π/6)/ 5(cos 2π/3 + i sin 2π/3)
=3 ( cos(5π/6-2π/3)+ isin(5π/6-2π/3) )
= 3 { cos (π/6) + i sin(π/6 ) }
= 3 ( √3
2 +i 1
2 )
= 3 √3
2 +i 3
2

5Pre-Calculus
Q4.
a)
6(cos 60˚ + i sin 60˚)
= 6( ½ + i √ 3
2 )
= 3 + i 3 √3
b)
8(cos 5π/6 + I sin 5π/6)
= 8 ( cos (π- π/6) + i sin (π- π/6) )
= 8 ( -cos (π/6) + i sin (π/6) )
= 8 ( - √ 3
2 + i ½ )
= -4 √3 + 4i .